Question

1. Consider the general form of the utility for goods that are
perfect complements.

a) Why won’t our equations for finding an interior solution to the
consumer’s problem work for this kind of utility? Draw(but do not
submit) a picture and explain why (4, 16) is the utility maximizing
point if the utility is U(x, y) = min(2x, y/2), the income is $52,
the price of x is $5 and the price of y is $2. From this picture
and explanation, can you describe where on the indifference curve
the utility maximizing point is for any budget for this utility? (1
point)

b) Use the last part of (a) help you figure out how to write an equation which will contain all the points you described. Using this equation and the budget in (a), solve for the utility maximizing bundle and show that (4, 16) turns out to be the solution. (1 point)

c) Repeat part (b) but now use a generic budget pxx + pyy = I instead of the specific prices and income to find the demands for good x*(px,py,I) and good y*(px,py,I) associated with the utility is U(x, y) = min(2x, y/2). Plug in the prices and income from

**

(b) to verify that the functions gives you the correct (x , y ) in
the one case where you

know the answer. (1 point)

2. A consumer is choosing between 3 goods, (x1, x2, x3). The
consumer’s utility is

1⁄4 1⁄2 1⁄4

?(?1,?2,?3) = ?1 ?2 ?3 . The consumer has I = 128 and faces prices
P1 = 2, P2 = 16

and P3 = 32.

a) Write out the Lagrangian for the consumers problem and find the
equations for the first order conditions to a solution. (1
point)

b) Use the first order conditions to verify that x1=16, x2=4 and x3=1 is the solution to the consumer’s problem. What is the additional utility from an extra $ for this consumer in this situation? (1 point)

12 3.GivenaU(x,y)=x3y3,Px =1,Py =2,andI=96:

a) Find the optimal (x,y) and the utility from that bundle. (1
point)

b) Find what happens when you change Px = 8 but keep Py = 2 two
different ways:

i) Keep the income = 96

ii) Keep the utility at the same level you found in (a)

Do these two methods give you the same answer? Does this make sense? Explain. (2 points)

c) How much does the answer to (b) part (ii) cost at the prices in part (b)? Is it more or less than 96? Does that make sense? (1 point)

d) For a general px, py, and I or U depending on the problem, find x * ( p x , p y , I ) and xc (px , py ,U). (Note: For this problem, don’t need either of the demands

for y) (1 point)

2

4. Answer the following based on a utility function U(x, y) = x3 +
y , P = 2 and P = 9 .

a) If income = 90, find the optimal quantity of the two goods. Now change income to 48 and try to solve this problem using the general method. Explain why the solution you get cannot be correct and find the actual solution to the problem. (note: a picture may be helpful but don’t submit it.) (1 point)

b) Find the demand functions for x*(px,py,I) and y*(px,py,I) assuming both x>0 and y>0. (1 point)

c) Explain from these functions why it will always be that x>0, but y>0 only if a condition is met. Show that condition. Also show the demand functions for x and y if the condition is not met. Explain how this result is consistent with part a. (1 point)

d) Find the indirect utility function for the case where y>0 and for where y=0. (1 point)

e) Find the minimum expenditure associated with px=2, py=9 and a utility of 13. Explain why this answer is consistent with earlier results. (1 point)

Answer #1

I have answered the first question for you. I hope it helps.

Answer 1:

(a)The Indifference curve for a perfect complement case comes in the form of an L-shaped curve. Due to this kink, the function is non-differentiable and hence, the use of lagrange is not fruitful since lagrange method requires you to differentiate the optimization function. That is why we can't find an interior solution using lagrange.

I have plotted the ICs for the given utility function and the budget line, 5x + 2y = 52, on a graph in the picture below to show that the optimum bundle is (4,16).

We plot the budget line using the Intercept-form of line equation as shown in the text picture above. The ICs for the given utility functions are the L -shaped level curves that will have kinks at the point 2x=y/2 => 4x=y.

The utility maximizing bundle is the one where the individual maximizes his/her utility function given the budget constraint, i.e. the point where the indifference curve is tangent to the budget line. In the picture above, that point is (4,16) where the IC1 (One IC out of the many from the family of ICs of the utility function U = min {2x,y/2}) is tangent to the budget line 5x + 2y = 52.

(b) Now, we have two equations, 4x=y ............(1)

5x + 2y = 52 ......................(2)

Substituting (1) in (2) gives us: 5x + 2*4x = 52 => 13x=52 => x*=4

And, y=4x => y=4*4 => y*=16

This gives us the utility maximizing bundle, (x*,y*) = (4,16).

(c) Now, we have two equations, 4x=y ..................(3)

Px*x + Py*y = I .................(4)

Substituting (3) in (4) gives us : Px*x + Py*4x = I => x (Px+4Py) = I

=> x (Px,Py,I) = I / (Px+4Py) {Demand function for x}

And, y (Px,Py,I) = 4* x = 4*I / (Px+4Py) {Demand function for y}

Now, for I=$52, Px=$5, and Py=$2, we get: x = 52/ (5+4*2) = 52/13 = 4

y = 4*52/(5+4*2) = 208/13 = 16

Thus, we get our utility optimizing bundle (4,16) back.

A consumer has utility function U(x, y) = x + 4y1/2 .
What is the consumer’s demand function for good x as a function of
prices px and py, and of income m, assuming a
corner solution?
Group of answer choices
a.x = (m – 3px)/px
b.x = m/px – 4px/py
c.x = m/px
d.x = 0

Consider a consumer with the following utility function: U(X,
Y ) = X1/2Y 1/2
(a) Derive the consumer’s marginal rate of substitution
(b) Calculate the derivative of the MRS with respect to
X.
(c) Is the utility function homogenous in X?
(d) Re-write the regular budget constraint as a function of PX
, X, PY , &I. In other words, solve the equation for Y .
(e) State the optimality condition that relates the marginal
rate of substi- tution to...

Jane’s utility function has the following form: U(x,y)=x^2
+2xy
The prices of x and y are px and py respectively. Jane’s income
is I.
(a) Find the Marshallian demands for x and y and the indirect
utility function.
(b) Without solving the cost minimization problem, recover the
Hicksian demands for x and y and the expenditure function from the
Marshallian demands and the indirect utility function.
(c) Write down the Slutsky equation determining the effect of a
change in px...

Consider a consumer with the utility function U(x, y) = min(3x,
5y). The prices of the two goods are Px = $5 and Py = $10, and the
consumer’s income is $220. Illustrate the indifference curves then
determine and illustrate on the graph the optimum consumption
basket. Comment on the types of goods x and y represent and on the
optimum solution.

8) Suppose a consumer’s utility function is defined by
u(x,y)=3x+y for every x≥0 and y≥0 and
the consumer’s initial endowment of wealth is w=100. Graphically
depict the income and
substitution effects for this consumer if initially Px=1 =Py and
then the price of commodity x
decreases to Px=1/2.

Consider a consumer whose utility function is
u(x, y) = x + y (perfect substitutes)
a. Assume the consumer has income $120 and initially faces the
prices px = $1 and py = $2. How much x and y would they buy?
b. Next, suppose the price of x were to increase to $4. How
much would they buy now?
c. Decompose the total effect of the price change on demand
for x into the substitution effect and the...

Suppose a consumer has the utility function U (x, y) = xy + x +
y. Recall that for this function the marginal utilities are given
by MUx(x,y) = y+1 and MUy(x,y) = x+1.
(a) What is the marginal rate of substitution MRSxy?
(b)If the prices for the goods are px =$2 and py =$4,and if the
income of the consumer is M = $18, then what is the consumer’s
optimal affordable bundle?
(c) What if instead the prices are...

A consumer's preferences are given by the utility function
u=(107)^2+2(x-5)y and the restrictions x>5 and y>0 are
imposed.
1. Write out the Lagrangian function to solve the consumer's
choice problem. Use the Lagrangian to derive the first order
conditions for the consumer's utility maximizing choice problem.
Consider only interior solutions. Show your work.
2. Derive the Optimal consumption bundles x*(px,py,w) and
y*(px,py,w)
3. Use the first order condition from 1 to calculate the
consumer's marginal utility of income when w=200,...

Ginger's utility function is U(x,y)=x2y with associated marginal
utility functions MUx=2xy and MUy=x2. She has income I=240 and
faces prices Px= $8 and Py =$2.
a. Determine Gingers optimal basket given these prices and her
in.
b. If the price of y increase to $8 and Ginger's income is
unchanged what must the price of x fall to in order for her to be
exactly as well as before the change in Py?

Tom spends all his $100 weekly income on two goods, X and Y. His
utility function is given by U(X,Y) =XY. If Px=4 and Py=10
a) how much of each good should the buy? Please show graphically
(specify clearly x-axis and y-axis and intercepts of each axis)
b) Same as Problem 1, except now Tom’s utility function is given
by U(X,Y) = X^1/2 Y^1/2
c) Find the MRS at the tangent points (optimal point) in a and
b.
Can you...

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